L(s) = 1 | + 4·9-s + 20·17-s − 2·25-s − 24·41-s − 6·49-s − 36·73-s − 6·81-s − 16·89-s − 48·97-s + 72·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 80·153-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | + 4/3·9-s + 4.85·17-s − 2/5·25-s − 3.74·41-s − 6/7·49-s − 4.21·73-s − 2/3·81-s − 1.69·89-s − 4.87·97-s + 6.77·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 6.46·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.208234946\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.208234946\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
good | 3 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 3 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 23 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.03597694554568477782925793613, −6.76524858234280497576853962935, −6.67451998530025654023698139397, −6.29917760001774489458982200150, −5.89138904798455870273502042289, −5.73709645719161143310994940025, −5.61312760757611456318205966009, −5.52161813586427681127493154043, −5.30005584770578324381022678625, −4.82318302801540567901277684336, −4.80729650171502837973667391370, −4.42154850135880767583686358960, −4.12950482803184405001146979328, −4.03421058638350864897326894513, −3.65612567421964293048161167803, −3.19805394882484495721313956110, −3.17598049585244861117592391403, −3.01287100473840157779367045948, −3.00870965464448832371541584128, −2.08225422170575501897262664898, −1.86826817566729217909127306176, −1.50571303898887487008178547615, −1.26258727654046097602948520640, −1.17055033005542009634975040100, −0.37043274595792950317024668851,
0.37043274595792950317024668851, 1.17055033005542009634975040100, 1.26258727654046097602948520640, 1.50571303898887487008178547615, 1.86826817566729217909127306176, 2.08225422170575501897262664898, 3.00870965464448832371541584128, 3.01287100473840157779367045948, 3.17598049585244861117592391403, 3.19805394882484495721313956110, 3.65612567421964293048161167803, 4.03421058638350864897326894513, 4.12950482803184405001146979328, 4.42154850135880767583686358960, 4.80729650171502837973667391370, 4.82318302801540567901277684336, 5.30005584770578324381022678625, 5.52161813586427681127493154043, 5.61312760757611456318205966009, 5.73709645719161143310994940025, 5.89138904798455870273502042289, 6.29917760001774489458982200150, 6.67451998530025654023698139397, 6.76524858234280497576853962935, 7.03597694554568477782925793613